Lemma 7.3.14. Let \(P\) be a set of primes and let \(X\) be a spectrum satisfying \(X[P^{-1}] = 0\) and \(X/p = 0\) for every \(p \in P\). Then \(X = 0\).
Proof. For every \(p\in P\), the condition \(X/p=0\) means that multiplication by \(p\) on \(X\) is an isomorphism. Thus \(X\) is \(P\)-divisible, hence \(\S [P^{-1}]\)-local by Lemma 7.2.8. On the other hand, the isomorphism \(X[P^{-1}] \cong X\otimes \S [P^{-1}]\) shows that \(X\) is \(\S [P^{-1}]\)-acyclic. It follows from Lemma 7.1.6 that \(X = 0\). □
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